Suppose that the function fis defined, for all real numbers, as follows. x-2 if x < -2 f(x) = 3x+2 if x2-2 Graph the function f. Then determine whether or not the function is continuous.
Suppose that the function fis defined, for all real numbers, as follows. x-2 if x < -2 f(x) = 3x+2 if x2-2 Graph the function f. Then determine whether or not the function is continuous.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Piecewise Function and Continuity
#### Definition of the Function
Suppose that the function \( f \) is defined, for all real numbers, as follows:
\[
f(x) = \begin{cases}
x - 2 & \text{if } x < -2 \\
3x + 2 & \text{if } x \geq -2
\end{cases}
\]
#### Instructions
Graph the function \( f \). Then determine whether or not the function is continuous.
#### Graphing the Function
To graph the piecewise function \( f \), follow these steps:
1. **For \( x < -2 \)**:
- The function is \( f(x) = x - 2 \).
- This is a linear function with a slope of 1 and a y-intercept of -2.
- Plot the line for \( x < -2 \).
2. **For \( x \geq -2 \)**:
- The function is \( f(x) = 3x + 2 \).
- This is a linear function with a slope of 3 and a y-intercept of 2.
- Plot the line for \( x \geq -2 \).
Remember to plot these sections on the same set of axes to create the complete piecewise graph.
#### Determining Continuity
After graphing the function, check the following to determine the continuity at \( x = -2 \):
- A function is continuous at a point if the limit as \( x \) approaches that point from both the left and the right is equal to the function's value at that point.
- Specifically, verify:
- \( \lim_{x \to -2^-} f(x) \)
- \( \lim_{x \to -2^+} f(x) \)
- \( f(-2) \)
#### Interactive Tools
To draw the function, you can use the tools provided:
- A pencil to draw the lines.
- A point to mark specific coordinates (e.g., checking the value at \( x = -2 \)).
#### Question
Is the function continuous?
**Options:**
- Yes
- No
Use the information gathered from the graph and continuity check to answer this question.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa15b1ffc-a64c-4923-946b-bd8a8c03fd77%2Fd0f62464-8e71-4f0a-a6e4-45e78fa3dd33%2Ftwn2aw_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Piecewise Function and Continuity
#### Definition of the Function
Suppose that the function \( f \) is defined, for all real numbers, as follows:
\[
f(x) = \begin{cases}
x - 2 & \text{if } x < -2 \\
3x + 2 & \text{if } x \geq -2
\end{cases}
\]
#### Instructions
Graph the function \( f \). Then determine whether or not the function is continuous.
#### Graphing the Function
To graph the piecewise function \( f \), follow these steps:
1. **For \( x < -2 \)**:
- The function is \( f(x) = x - 2 \).
- This is a linear function with a slope of 1 and a y-intercept of -2.
- Plot the line for \( x < -2 \).
2. **For \( x \geq -2 \)**:
- The function is \( f(x) = 3x + 2 \).
- This is a linear function with a slope of 3 and a y-intercept of 2.
- Plot the line for \( x \geq -2 \).
Remember to plot these sections on the same set of axes to create the complete piecewise graph.
#### Determining Continuity
After graphing the function, check the following to determine the continuity at \( x = -2 \):
- A function is continuous at a point if the limit as \( x \) approaches that point from both the left and the right is equal to the function's value at that point.
- Specifically, verify:
- \( \lim_{x \to -2^-} f(x) \)
- \( \lim_{x \to -2^+} f(x) \)
- \( f(-2) \)
#### Interactive Tools
To draw the function, you can use the tools provided:
- A pencil to draw the lines.
- A point to mark specific coordinates (e.g., checking the value at \( x = -2 \)).
#### Question
Is the function continuous?
**Options:**
- Yes
- No
Use the information gathered from the graph and continuity check to answer this question.
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